\subsection*{2012-06-24}
\begin{enumerate}
  \item Nathan 

  \item We wish to obtain a scalar action that gives rise to Einstein field equations through variational methods. 
  Let's figure out what equation varying of the Einstein-Hilbert action gives.
  \begin{equation}
    S_{EH} = \tfrac{1}{2 \kappa} \int_M d^nx \sqrt{-g} R
  \end{equation}
  In varying $S_{EH}$ we impose $\delta g_{\alpha\beta} |_{\partial M} = 0$. Variation of $S_{EH}$ gives: 
  \begin{equation}
    \delta S_{EH} = \int_M d^n x \sqrt{-g} \left( R_{\alpha\beta} - \tfrac{1}{2} R g_{\alpha \beta} \right) \delta g^{\alpha \beta} + \int_M d^n x \sqrt{-g} \nabla_\sigma \left( g^{\alpha \beta} \delta \Gamma^\sigma_{\beta \alpha} - g^{\alpha \sigma} \delta \Gamma^\gamma_{\alpha \gamma} \right)
  \end{equation}
  In deriving of the above equation we only used a couple of handy identities: $\delta g_{\alpha \beta} = -g_{\alpha \mu} g_{\beta \nu} \delta g^{\mu \nu}$, 
\end{enumerate}
